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Calculus with Fuzzy Numbers

Witold Kosinski1, Piotr Prokopowicz2,3, and Dominik Slezak4,1

1 Polish-Japanese Institute of Information Technology,Research Center, ul. Koszykowa 86, 02-008 Warsaw, Poland

[email protected] Institute of Fundamental Technological Research, PAS (IPPT PAN),

PSWiP, ul.Swietokrzyska 21, 00-049 Warsaw, Poland3 University of Bydgoszcz Institute of Environmental Mechanics

and Applied Computer Science,ul. Chodkiewicza 30, 85-064 Bydgoszcz, Poland

[email protected] The University of Regina, Department of Computer Science,

Regina, SK, S4S 0A2 [email protected]

Abstract. Algebra of ordered fuzzy numbers (OFN) is defined to handle withfuzzy inputs in a quantitative way, exactly in the same way as with real numbers.Additional two structures: algebraic and normed (topological) are introduced todefine a general form of defuzzyfication operators. A useful implementation of aFuzzy Calculator allows counting with the general type membership relations.

1 Introduction

Nowadays, fuzzy approach is helpful while dealing with non-exact data involving hu-man vagueness in large multimedia databases. In real-life problems both parametersand data used in mathematical modelling are vague. The vagueness can be described byfuzzy numbers and sets.

Communication, data mining, pattern recognition, system modelling, diagnosis, im-age analysis, fault detection and others are fields where clustering plays an importantrole. However, in practice constructed clusters overlap, and some data vectors belongto several clusters with different degrees of membership. A natural way to describe thissituation results in implementing the fuzzy set theory [18,21], where the membershipof a vector xk to the i-th cluster Ui is a value from the unit interval [0, 1]. Approximatereasoning, on the other hand, by means of compositional fuzzy rules of inference canhelp in dealing with uncertainty inherent in the processed knowledge, especially whenbuilding a fuzzy decision support system for decision making tasks in different domainsof applications [25,27,28].

All those situations require well known fuzzy logic and even more – arithmetics offuzzy numbers, in order to perform operations on fuzzy observations. Fuzzy conceptshave been introduced in order to model such vague terms as observed values of somephysical or economic terms, like pressure values or stock market rates, that can beinaccurate, can be noisy or can be difficult to measure with an appropriate precision

L. Bolc et al. (Eds.): IMTCI 2004, LNAI 3490, pp. 21–28, 2005.c© Springer-Verlag Berlin Heidelberg 2005

22 W. Kosinski, P. Prokopowicz, and D. Slezak

because of technical reasons. In our daily life there are many cases where observationsof objects in a population are fuzzy. In modern complex and large-scale systems it isdifficult to adopt the systems using only exact data. Also in this case, it is inevitable toadopt non-exact data involving human vagueness. In this way we are approaching theconcept of the fuzzy observation.

The commonly accepted theory of fuzzy numbers [1] is that set up in [4] by Duboisand Prade in 1978, who proposed a restricted class of membership functions, called(L, R)–numbers with two so-called shape functions: L and R. (L, R)–numbers canbe used for the formalisation of basic vague terms. However, approximations of fuzzyfunctions and operations are needed if one wants to follow the Zadeh’s extension prin-ciple [26,27]. It leads to some drawbacks that concern properties of fuzzy algebraicoperations, as well as to unexpected and uncontrollable results of repeatedly appliedoperations [23,24].

Classical fuzzy numbers (sets) are convenient as far as a simple interpretation inthe set-theoretical language is concerned. However, we could ask: How can we imaginefuzzy information, say X , in such a way that by adding it to the fuzzy number A thefuzzy number C will be obtained? In our previous papers (see [14] for references) wetried to answer that question in terms of so-called improper parts of fuzzy numbers. Inthis paper we consider the algebra of ordered fuzzy numbers that leads to an efficienttool in dealing with unprecise, fuzzy quantitative terms.

2 Motivations

One of the goals of our paper is to construct a revised concept of a fuzzy number, andat the same time to have the algebra of crisp (non-fuzzy) numbers inside the concept.The other goal is to preserve as much of the properties of the classical so-called crispreals R as possible, in order to facilitate real world applications as e.g. in fuzzy controlsystems. The new concept allows for utilising the fuzzy arithmetic and constructing analgebra of fuzzy numbers. By doing this, the new model of fuzzy numbers has obtainedan extra feature which was not present in the previous ones: neither in the classicalZadeh’s model, nor in the more recent model of so-called convex fuzzy numbers. Thisfeature, called in [12,14] the orientation, requires a new interpretation as well as aspecial care in dealing with ordered fuzzy numbers. To avoid confusion at this stage ofdevelopment, let us stress that any fuzzy number, either classical (crisp or convex fuzzy)or ordered (new type), has its opposite number which is obtained from the given numberby multiplication with minus one. For the new type of fuzzy numbers, multiplicationby a negative real not only affects the support, but also the orientation swaps. It isimportant that to a given ordered fuzzy number two kinds of opposite elements aredefined: the classical, one can say an algebraic opposite number (element) obtainedby its multiplication with a negative crisp one, and the complementary number whichdiffers from the opposite one by the orientation. Relating to an ordered fuzzy number,its opposite and complementary elements make the calculation more complex, howeverwith new features. On the one hand a sum of an ordered fuzzy number and its algebraicopposite gives a crisp zero, like in the standard algebra of real number. On the otherhand the complementary number can play the role of the opposite number in the sense

Calculus with Fuzzy Numbers 23

of the Zadeh’s model, since the sum of the both – the (ordered fuzzy) number and itscomplementary one – gives a fuzzy zero, non-crisp, in general.

We have to admit that the application of the new type of fuzzy numbers is restrictedto such real-life situations where also the modelled circumstances provide informationabout orientation. In particular, in majority of existing approaches, for a fuzzy numberA the difference A−A gives a fuzzy zero. However, this leads to unbounded growth ofthe support of fuzziness if a sequence of arithmetic operations is performed between two(classical) fuzzy numbers. To overcome this unpleasant circumstance the concept of theorientation of a fuzzy number has been introduced as well as simple operations betweenthose new objects, called here ordered fuzzy numbers, which are represented by pairs ofcontinuous functions defined on the unit interval [0,1]. Those pairs are the counterpartsof the inverses of the increasing and decreasing parts of convex fuzzy numbers. In aparticular case, for the pairs (f, g) where f, g ∈ C0([0, 1]) which satisfy: 1)f ≤ gand 2)f and g are invertible, with f increasing and g decreasing, one can recover theclass of fuzzy numbers called convex ones [3,20]. Then as long as multiplication bynegative numbers is not performed, classical fuzzy calculus is equivalent to the presentoperations defined for ordered fuzzy numbers (with negative orientation).

Doing the present development, we would like to refer to one of the very first repre-sentations of a fuzzy set defined on a universe X (the real axis R, say) of discourse. Inthat representation (cf. [6,26]) a fuzzy set (read here: a fuzzy number) A is defined as aset of ordered pairs {(x, µx); x ∈ X}, where µx ∈ [0, 1] has been called the grade (orlevel) of membership of x in A. At that stage, no other assumptions concerning µx havebeen made. Later on, one assumed that µx is (or must be) a function of x. However,originally, A was just a relation in a product space X × [0, 1].

3 Attempts

A number of attempts to introduce non-standard operations on fuzzy numbers havebeen made [1,3,7,22,23]. It was noticed that in order to construct operations more suit-able for their algorithmisation a kind of invertibility of their membership functions isrequired. In [10,16,17,20] the idea of modelling fuzzy numbers by means of convexor quasi-convex functions (cf. [19]) is discussed. We continue this work by definingquasi-convex functions related to fuzzy numbers in a more general fashion (called afuzzy observation, compare its definition in [14]) enabling modelling both dynamics ofchanges of fuzzy membership levels and the domain of fuzzy real itself. It is worth-while to mention here that even starting from the most popular trapezoidal membershipfunctions, algebraic operations can lead outside this family, towards such generalisedquasi-convex functions.

That more general definition enables to cope with several drawbacks. Moreover, itseems to provide a solution for other problems, like, e.g., the problem of defining anordering over fuzzy numbers (cf.[12]). Here we should mention that Klir was the firstwho in [7] has revised fuzzy arithmetic to take relevant requisite constraint (the equalityconstraint, exactly) into account and obtained A − A = 0, with crisp 0, as well as theexistence of inverse fuzzy numbers for the arithmetic operations. Some partial resultsof similar importance were obtained by Sanchez in [22] by introducing an extended


operation of a very complex structure. Our approach, however, is much simpler frommathematical point of view, since it does not use the extension principle but refers tothe functional representation of fuzzy numbers in a more direct way.

In the classical approach the extension principle gives a formal apparatus to carryover operations (e.g. arithmetic or algebraic) from sets to fuzzy sets. Then in the case ofthe so-called convex fuzzy numbers (cf. [3]) the arithmetic operations are algorithmisedwith the help of the so-called α-sections of membership functions. It leads to the opera-tions on intervals. The local invertibility of quasi-concave membership functions, on theother hand, enables to define operations in terms of the inverses of the correspondingmonotonic parts, as was pointed out in our previous papers [11,13]. In our last paper[14] we went further and have defined a more general class of fuzzy number, calledordered fuzzy number, just as a pair of continuous functions defined on the interval[0, 1]. Those pairs are counterparts of the mentioned inverses.

4 Ordered Fuzzy Numbers

Here the concept of membership functions [1] is weakened by requiring a mere mem-bership relation and a fuzzy number is identified with an ordered pair of continuousreal functions defined on the interval [0, 1].

Definition 1. By an ordered fuzzy number A we mean an ordered pair A = (f, g) ofcontinuous functions f, g : [0, 1] → R.

We call the corresponding elements: f – the up-part and g – the down-part of the fuzzynumber A. The continuity of both parts implies their images are bounded intervals, say

Fig. 1. a) Ordered fuzzy number, b) Ordered fuzzy number presented as fuzzy number in classicalmeaning, c) Simplified mark denotes the order of inverted functions


UP and DOWN , respectively (Fig. 1a). Let us use symbols to mark boundaries forUP = (lA, 1+

A) and for DOWN = (1−A, pA).In general, functions f, g need not be invertible, only continuity is required; they

give the real variable x ∈ R in terms of y ∈ [0, 1], if one refers to the classical mem-bership function denotation. If we add a function of x on the interval [1+

A, 1−A] withconstant value equal to 1, we may define a kind of membership function (relation) of afuzzy set. When f ≤ g are both invertible, f is increasing, and g is decreasing, we get amathematical object which presents a convex fuzzy number in the classical sense [7,23].

We can appoint an extra feature, named an orientation (marked by an arrow inFig. 1c), to underline that we are dealing with an ordered pair of functions.

Definition 2. Let A = (fA, gA), B = (fB, gB) and C = (fC , gC) be mathematicalobjects called ordered fuzzy numbers. The sum C = A + B, subtraction C = A − B,product C = A · B, and division C = A/B are defined by formula

fC(y) = fA(y) � fB(y) ∧ gC(y) = gA(y) � gB(y) (1)

where "�" works for "+", "−", "·", and "/", respectively, and where A/B is defined, iffzero does not belong to intervals UP and DOWN of B.

Subtraction of B is the same as addition of the opposite of B, i.e. the number (−1) ·B.If for A = (f, g) we define its complement A = (−g,−f) (please note that A �=(−1) · A), then the sum A + A gives a fuzzy zero 0 = (f − g,−(f − g)) in the senseof the classical fuzzy number calculus.

Definition 3. Let A = (fA, gA), B = (fB, gB) and C = (fC , gC) be mathematicalobjects called ordered fuzzy numbers. The maximum C = max(A, B) = A ∨ B andthe minimum C = min(A, B) = A ∧ B are defined by formula

fC(y) = func {fA(y), fB(y)} ∧ gC(y) = func {gA(y), gB(y)} (2)

where "func" works for "max" and "min", respectively.

Many operations can be defined in this way, suitable for the pairs of functions. TheFuzzy Calculator has been already created as a calculation tool by our co-workerMr. Roman Kolesnik [9]. It lets an easy future use of all mathematical objects describedas ordered fuzzy numbers.

5 Further Extensions

Banach Algebra. The pointwise multiplication by a scalar (crisp) number, togetherwith the operation addition lead to a linear structure R the set of all OFN’s, which isisomorphic to the linear space of real 2D vector-valued functions defined on the unitinterval I = [0, 1]. Further, one can introduce the norm over R as follows:

||x|| = max(sups∈I

|xup(s)|, sups∈I

|xdown(s)|) (3)


Hence R can be identified with C([0, 1]) × C([0, 1]). Finally, R is a Banach algebrawith the unity (1†, 1†)– a pair of constant functions 1†(y) = 1, for y ∈ [0, 1]. Previ-ously, a Banach structure of an extension of convex fuzzy numbers was introduced byGoetschel and Voxman [5]. However, they were only interested in the linear structureof this extension.

Order Relation and Ideals. On the space R we can introduce a pre–order [15] bydefining a function W with the help of the relation

W (A) = (xup + xdown), (4)

its value W (A) is a variation of the number A = (xup, xdown). Then we say that theordered fuzzy number A is not smaller than the number B, and write A � B, if

W (A) ≥ W (B) ⇔ W (A − B) ≥ 0 (5)

i.e. when the function W (A − B) is non–negative. We say that the number C is non-negative if its variation is not smaller than zero, i.e. when W (C) ≥ 0. Notice that thereare ordered fuzzy numbers that are not comparable with zero. We say that a number Dis around zero if its variation is the constant function equal to zero, i.e. W (D) = 0†.Thanks to this relation in the Banach algebra R we may define two ideals: the left andthe right ones, which are non-trivial and possess proper divisors of zero [15].

Defuzzyfication. This is the main operation in fuzzy inference systems and fuzzy con-trollers [1,17,25]. The problem arises what can be done for the generalisation of clas-sical fuzzy numbers onto ordered fuzzy numbers? In the case of the product space R,according to the Banach-Kakutami-Riesz representation theorem, each bounded linearfunctional φ is given by a sum of two bounded, linear functionals defined on the factorspace C([0, 1]), i.e.

φ(xup, xdown) =∫ 1

0

xup(s)µ1(ds) +∫ 1

0

xdown(s)µ2(ds) (6)

where the pair of continuous functions (xup, xdown) ∈ R represents an ordered fuzzynumber, and µ1, µ2 are two Radon measures on [0, 1].

From this formula an infinite number of defuzzyfication methods can be defined. Inparticular, the standard procedure given in terms of the area under membership functioncan be generalised. It is realised by the pair of linear combinations of the Lebesguemeasure of [0, 1]. Moreover, a number of non-linear defuzzyfication operators can bedefined as compositions of multivariant nonlinear functions defined on the Cartesianproducts of R and linear continuous functionals on the Banach space R [15].

It is worthwhile to mention that some further generalisations of ordered fuzzy num-bers to ordered fuzzy sets (defined on a different universe than reals R) can be intro-duced (cf. [15]). Moreover, one can think about weakening of the continuity assumptionmade in our fundamental definition of ordered fuzzy numbers and to consider pairs ofreal valued functions of the interval [0, 1] that are of bounded variation. Then all al-gebraic properties of new objects will be preserved with a small change of the norm.However, it will the subject of the next publication.


Acknowledgement. The research work on the paper was partially done in the frame-work of the KBN Project (State Committee for Scientific Research) No. 4 T11C 038 25.The third author was partially supported by the research grant of the Natural Sciencesand Engineering Research Council of Canada.

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